Complexity of greedy edge-colouring

The Grundy index of a graph G =(V, E) is the greatest number of colours that the greedy edge-colouring algorithm can use on G. We prove that the problem of determining the Grundy index of a graph G=(V, E) is NP-hard for general graphs. We also show that this problem is polynomial-time solvable for caterpillars. More specifically, we prove that the Grundy index of a caterpillar is Δ(G) or Δ(G)+1 and present a polynomial-time algorithm to determine it exactly.